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# Time Series Analysis

The sequence of the observation ordered in time has called time series. We can display best time series by using scatter plot. In the scatter plot on y-axis or vertical axis we plotted the time series values x and the horizontal axis is plotted time t. Here time is independent variable. We collect these observations at equally spaced and discrete time intervals. Time series can be decomposed into three components like the trend, the seasonal and the irregular. An important step in analyzing to time series data is to consider the pattern of statics data, According to patterns of data the models can be utilized. Let me explain types of time series components can be distinguished. Like that – First we talk about horizontal time series, when data values are fluctuating around a constant value is called horizontal time series.

If we have a long term increase and decrease in the data is called a trend. Seasonal factor is a systematic calendar bases movement. Now we talk about a fourth component of cyclic time series. If the data exhibit rises and falls that are not fixed point. The preceding patterns are a combination of the many data series. When we separate out the existing pattern in any time series data, the remains unidentifiable pattern, form the error or random component. The data plotted over time and data plotted against individual seasons in which data are observed, these patterns help exploring data. An additive or multiplicative model – Yt = Tt + St + Et or Yt = Tt . St . Et, Here Yt = Original time series data, Tt = Trend component, St = Seasonal component, Et = Error / irregular component. Let we talk about moving average and exponential smoothing method. First we talk about simple moving average: - A numerical average of the last N data points is called a moving average.

There are prior moving average, centered moving average etc. in the time series literature. Moving average at time t and taken over N periods, is given by Mt[1] = (Yt + Yt-1 + Yt – 2 + Yt 3 + …………+ Yt-n+1 ) / N. Here Yt is called the observe response at time t. Here we divided sum of t – N + 1 term by N. We can state that by another way like Mt[1] = Mt - 1[1] + (Yt – Yt-n) / N. Exponentially weighted moving average. Simple Exponential smoothing (SES): - Let the time series data be denoted by Y1, Y2, Y3 Y4,…………,Yt. Suppose the next value is Yt+1 that is yet observed with forecast for itdenotedby Ft. Then the forecast Ft+1 is based on the most recent observation Yt with weight value x and the most recent forecast Ft with a weight of (1 – x), here x is smoothing constant between 0 and 1. Forecast for period t + 1 is given by Ft+1 = Ft + x(Yt - Ft )

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